number theory - What is the remainder when $3^{57} + 27$ is divided by $28$?

Problem: What is the remainder when $$3^{57}+27$$ is divided by $28$ ?

Source: I'm pretty much interested in calculus (you can refer to my previous posts) but I have to prepare for a test where they even put up problems on elementary number theory. I got this problem from a practice set and it stumped me. I looked up for similar questions on the website and most of them include the use of $\mathrm{mod}$. I don't know what it is, and I haven't got time to understand it as I also have to deal with physics and chemistry at the same time. I have solved a very few problems of this kind (mainly divisibility) using mathematical induction and binomial theorem last year.

My try: When you got integral calculus embedded into your mind, how do you approach without using it? I have tried to develop a function:
$$f(x) = \int(a^x+b)\mathrm{d}x$$
$$= \int{a^x}\mathrm{d}x + b\int \mathrm{d}x$$
$$= \frac{ a^{x+1}}{x+1} + bx + C$$
put limits $l_l = 0$ and $l_u = 57$ where $l_l$ and $l_u$ are lower and upper limits respectively.

But I have tried to solve it for no good. I can't think of a possible way, and my professor is unwilling to help me with it (duh!). I'm stuck. I have to perform better. So can you please give me an approach without using the $\mathrm{mod}$ function? All help appreciated!

Answer

You want remainder when $3^{57}+27$ is divided by $28$. Note that $3^{57}=(3^3)^{19}$.

$$3^{57}+27=(3^3)^{19}+27=(28-1)^{19}+27={19\choose0}28^{19}-{19\choose 1}28^{18}\cdot\cdot\cdot\cdot\cdot+{19\choose18}28-{19\choose19}+27=28k-1+27=28k+26$$

When divided by $28$, $28k+26$ gives $26$ as remainder.